Real algebraic curves with large finite number of real points

Brugallé, Erwan; Degtyarev, Alex; Itenberg, Ilia; Mangolte, Frédéric

doi:10.1007/s40879-019-00324-9

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Real algebraic curves with large finite number of real points

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Degtyarev, Alex
Max Planck Institute for Mathematics, Max Planck Society;

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Itenberg, Ilia
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s40879-019-00324-9
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arXiv:1807.03992.pdf
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Citation

Brugallé, E., Degtyarev, A., Itenberg, I., & Mangolte, F. (2019). Real algebraic curves with large finite number of real points. European Journal of Mathematics, 5(3), 686-711. doi:10.1007/s40879-019-00324-9.

Cite as: https://hdl.handle.net/21.11116/0000-0004-A656-0

Abstract

We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to
optimal curves of small degree. Our upper bound is sharp if the genus is small as compared to the degree. Some of the results are extended to other real algebraic surfaces, most notably ruled.